Definitions
Formally, each of the following definitions defines a concrete category, and every pair of these categories can be shown to be concretely isomorphic. This means that for every pair of categories defined below, there is an isomorphism of categories, for which corresponding objects have the same underlying set and corresponding morphisms are identical as set functions.
To actually establish the concrete isomorphisms is more tedious than illuminating. The simplest approach is probably to construct pairs of inverse concrete isomorphisms between each category and the category of topological spaces Top. This would involve the following:
- Defining inverse object functions, checking that they are inverse, and checking that corresponding objects have the same underlying set.
- Checking that a set function is "continuous" (i.e., a morphism) in the given category if and only if it is continuous (a morphism) in Top.
Read more about this topic: Characterizations Of The Category Of Topological Spaces
Famous quotes containing the word definitions:
“Lord Byron is an exceedingly interesting person, and as such is it not to be regretted that he is a slave to the vilest and most vulgar prejudices, and as mad as the winds?
There have been many definitions of beauty in art. What is it? Beauty is what the untrained eyes consider abominable.”
—Edmond De Goncourt (1822–1896)
“What I do not like about our definitions of genius is that there is in them nothing of the day of judgment, nothing of resounding through eternity and nothing of the footsteps of the Almighty.”
—G.C. (Georg Christoph)